It needs a back.
That would do it. But I don't want one (don't want the weight, the expense, or the block for a fairly beautiful wall), so I am hoping for another solution.
Can you entertain a diagonal brace? You need something that is neither horizontal or vertical.
Sure. Can you or someone advise me on how to put that on so that it is straight, as opposed to simply preventing it from deviating from its current crookedness?
(And making the next bookshelf start straight? I've got 3 to go!)
Given its current structure, you should be able to force it into a square shape.
Put one nail (only one) in diagonal brace, force to square, then put a second nail on on the other end of the brace. It should now be stable and then you can finish securing the brace. Just use a carpenters square to make sure you are dead on.
I can't see the other side in the picture. I assume the two vertical sides are parallel? If not, you have another problem and need to check the lengths of the shelves to make sure they are all the same.
The back would help, but as I have seen in a previous comment you don't want one.
My thoughts on your arguments against the backing:
cost and weight - You can get a couple of inexpensive and nice looking pieces of sheeting for like 4 bucks a pop and they are light as hell.
Look - You want to preserve being able to see the wall, but won't this be filled with books anyways, which would block viewing of the wall?
That being said, if you are surely against the backing, I would say, put a strip of straight lumber (1-2" wide) and afix it to back of both sides, screwing into the sides and the shelves themselves.
I think that would straighten it out a bit and would allow you to see the wall behind the shelf.
Sheeting: you mean plywood? Something else? I may go with Kit's suggestion below and just use it behind the bottom shelf.
Wood paneling rather.
My brother and I made a book case and used that. We had the same problem you are having and that fixed it right up. We did have to paint it though, it wouldn't have look good otherwise.
One option that would give you the strength, stability, and squareness you want would be to put a back on just the bottom shelf or two. That gives you the structural benefits of a back without covering the wall anywhere near eye level.
Hey, I like that. What would you use? How thick?
Hardboard should suffice.
I agree with GE - 1/4" hardboard would be ideal.